The Sum of Lagrange Numbers Jonah Gaster, Brice Loustau

The sum of Lagrange numbers Jonah Gaster, Brice Loustau

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Jonah Gaster, Brice Loustau. The sum of Lagrange numbers. 2020. hal-02961570

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JONAH GASTER AND BRICE LOUSTAU

Abstract. Combining McShane’s identity on a hyperbolic punctured torus with Schmutz’s work on the Markov Uniqueness Conjecture (MUC), we ﬁnd that MUC is equivalent to the identity ∞ X √ (3 − Ln) = 4 − ϕ − 2 n=1 √ 1+ 5 where Ln is the nth Lagrange number and ϕ = 2 is the golden ratio.

1. Preliminaries √ √ ∞ 1.1. Lagrange and Markov numbers. The Lagrange numbers L = {Ln}n=1 = { 5, 8,...} are a sequence of real numbers that naturally arise in Diophantine approximation. Hurwitz’s theorem states that for any irrational number x, there exists a sequence of rationals pn/qn converging to x with √ pn √1 x − q < 2 . In this expression, 5 is optimal, as can be shown by taking x = ϕ (the golden ratio). n 5qn √ It turns out that√ when x = ϕ and related numbers are√ excluded, 8 is the new best constant. By deﬁnition, L1 = 5 is the ﬁrst Lagrange number, L2 = 8 is the second Lagrange number, etc. ∞ The Markov numbers M = {mn}n=1 = {1, 2, 5, 13,...} are the positive integers that appear in a 3 Markov triple, i.e. a solution (x, y, z) ∈ Z to the cubic (1) x2 + y2 + z2 = 3xyz .

In 1880, Markov [ Mar79 , Mar80 ] discovered a remarkable connection between this cubic and the theory of binary quadratic forms, and proved the unexpected relation between Markov and Lagrange numbers: s 4 (2) Ln = 9 − 2 . mn Using the Vieta involution (x, y, z) 7→ (x, y, 3xy − z), it is easy to see that for any Markov number m, one can always ﬁnd a Markov triple (x, y, z = m) with 0 < x 6 y 6 z. The Markov Uniqueness Conjecture (MUC) asserts that such a triple is always unique. MUC was initially oﬀered by Frobenius in

1913 [ Fro13 ] and is notoriously diﬃcult [ Guy83 ]. For more context and detail, we refer to [ Aig15 , CF89 ].

1.2. The sum of Lagrange numbers. It is clear from ( 2 ) that Ln is an increasing sequence of positive 2 numbers that converges to 3 when n → +∞. Moreover, we have 3 − Ln ∼ 3m2 , and since mn > n P∞ n (actually mn is much greater, see § 3 ), the series n=1(3 − Ln) is convergent. In this paper, we prove: Theorem 1.1. The Markov Uniqueness Conjecture holds if and only if ∞ X √ (3) (3 − Ln) = 4 − ϕ − 2 . n=1 The proof is easily derived from the McShane identity on a hyperbolic punctured torus and a result of Schmutz regarding the well-known relationship between hyperbolic geometry and Markov numbers. It is nonetheless a striking identity, and could optimistically open a new path towards probing MUC.

Date: August 19, 2020. 1 2 JONAH GASTER AND BRICE LOUSTAU

Remark 1.2. Several authors have explored similar ideas, for instance [ Bow96 ], [ LT07 , §4.3].

Remark 1.3. Numerical computation conﬁrms the identity ( 3 ) convincingly, as we shall see in § 3 . This is not surprising since MUC has also directly been checked by computers for high values of n.

1.3. Markov numbers and the modular torus. The beautiful relationship between Markov numbers ∗ and hyperbolic geometry was discovered by Gorshkov [ Gor81 ] and Cohn [ Coh55 ]. Let T denote the once-punctured torus, i.e. the topological surface obtained by removing a point from the torus T 2. For a certain hyperbolic metric on T ∗, the lengths of simple closed geodesics on T ∗ are given by the Markov

numbers. We brieﬂy explain this connection and refer to e.g. [ Ser85 ] for more discussion. The character variety of the once-punctured torus is the cubic surface X deﬁned by the equation

(4) x2 + y2 + z2 = xyz .

∗ ∗ Hyperbolic metrics on T with ﬁnite volume correspond to real points of X . Indeed, let π1(T ) = ha, bi 2 2 ∗ where a and b are the standard generators of π1(T ) ≈ Z . Hyperbolic structures on T are parametrized ∗ by x = tr(A), y = tr(B), z = tr(AB) where A, B ∈ SL2(R) are (lifts of) the holonomies of a, b ∈ π1(T ). The condition that the metric has ﬁnite volume amounts to the peripheral curve aba−1b−1 having −1 −1 parabolic holonomy, i.e. tr(ABA B ) = −2. Using the classical trace relations in SL2(R), this 2 2 2 equation is rewritten x +y +z = xyz. We refer to e.g. [ Gol03 ] for more details on this correspondence. 3 The integer solutions (x, y, z) ∈ Z of ( 4 ) are clearly in bijection with Markov triples: x, y, z must all

be divisible by 3, and the reduced triple (x/3, y/3, z/3) veriﬁes ( 1 ). Thus Markov triples are the integral points of X (up to 1/3). In fact, the mapping class group Mod(T ∗) acts transitively on such triples, i.e. all corresponding hyperbolic tori are isometric. This hyperbolic torus is called the modular torus X, a 6-fold cover of the modular orbifold. Markov numbers can alternatively be described as one third of traces of simple closed geodesics on X:

3 M = {3 mn, n ∈ N} = {τ(γ), γ ∈ S} where we denote S the set of simple closed geodesics on X and τ(γ) the trace of the holonomy of γ ∈ S. It is natural to ask whether for any m ∈ M, the geodesic γ such that τ(γ) = 3m is unique up to an

isometry of X. It was proved by Schmutz [ Sch96 ] that this statement is equivalent to MUC.

1.4. Acknowledgments. We thank Ser Peow Tan and David Dumas for valuable feedback.

2. Proof of the theorem Greg McShane showed that, for any ﬁnite-volume hyperbolic metric on the punctured torus T ∗, X 1 1 = 1 + e`(γ) 2 γ∈S

where S is the set of simple closed geodesics and `(γ) indicates the length of γ [ McS98 ]. Recalling that the trace and length of γ are related by τ(γ) = 2 cosh(`(γ)/2), McShane’s identity can be rewritten X 2 X (5) 1 = = e−`(γ)/2 sech(`(γ)/2) 1 + e`(γ) γ γ s X 2 2 X 4 = · = 1 − 1 − . p 2 τ(γ) τ(γ)2 γ τ(γ) + τ(γ) − 4 γ THE SUM OF LAGRANGE NUMBERS 3

When T ∗ with its hyperbolic metric is chosen to be the modular torus X, let us denote m(γ) := τ(γ)/3 q 4 the associated Markov number (see § 1.3 ) and L(γ) := 9 − m(γ)2 the associated Lagrange number.

Reworking ( 5 ), McShane’s identity on the modular torus is simply rewritten: X (6) (3 − L(γ)) = 3 . γ∈S It remains to investigate the ﬁbers of the map γ 7→ L(γ) from simple closed geodesics on X to Lagrange numbers. It is not hard to show that all ﬁbers are nonempty: this is because Vieta involutions

act transitively on the Markov tree, and act as mapping classes on S. By Schmutz’s theorem [ Sch96 ], MUC is equivalent to each ﬁber of γ 7→ L(γ) being the Aut(X)-orbit of a single simple closed geodesic

on X. To ﬁnish the proof of Theorem 1.1 , we just need to count the number of elements of each orbit.

Lemma 2.1. Let S0 ⊂ S indicate the six shortest geodesics on X, and let S1 = S − S0. Each orbit Aut(X) y S0 has three elements, and each orbit of Aut(X) y S1 has six elements.

Proof. There is an Aut(X)-equivariant correspondence of S with lines in H := H1(X, Z). The stan- ∗ 2 dard generators a, b of π1(X) ≈ π1(T ) (as in § 1.3 ) provide a basis of H ≈ Z . The image of the homomorphism Aut(X) → PGL(2, Z) is the dihedral group with six elements, generated by 0 1 0 1 r = and σ = . −1 −1 1 0 1 The actions of r and σ on P H have ﬁxed points Fix(r) = ∅ and Fix(σ) = {[1 : 1], [1 : −1]}. This implies that all simple closed geodesics on X have six images under the action of Aut(X), except for the two geodesics corresponding to ab and ab−1, which have three such images apiece. These six geodesics are precisely the six shortest geodesics on X.

Let us now prove Theorem 1.1 , in fact the slightly more precise version: ∞ X √ Theorem 2.2. We have (3 − Ln) 6 4 − ϕ − 2, with equality if and only if MUC holds. n=1 Proof. Recall that X denotes the modular torus and S the set of simple closed geodesics on X. Let

S/Aut(X) indicate the set of Aut(X)-orbits in S. By ( 6 ), the McShane identity on X is rewritten: X X X (3 − L(γ)) = 3 − L(γ) = 3 . γ∈S A∈S/Aut(X) γ∈A

By Lemma 2.1 , the map γ 7→ L(γ) is 6-to-1 for γ ∈ S1 and 3-to-1 for γ ∈ S0. Therefore, we get   X X 6 + 3  (3 − L(γ)) = 3 . [γ]∈S1/Aut(X) [γ]∈S0/Aut(X) The six curves in S0 are the shortest geodesics√ in S, so√ the two Lagrange numbers they determine are the two smallest Lagrange numbers L1 = 5 and L2 = 8. The previous equality can be written   X 6 (3 − L(γ)) − 3 (3 − L1) + (3 − L2) = 3 , [γ]∈S/Aut(X) which we rewrite: X √ (3 − L(γ)) = 4 − ϕ − 2 . [γ]∈S/Aut(X)

The map [γ] 7→ L(γ) from S/Aut(X) to the set of Lagrange numbers L = {Ln, n ∈ N} is onto, and one-to-one if and only if MUC holds (see discussion above Lemma 2.1 ). The conclusion follows. 4 JONAH GASTER AND BRICE LOUSTAU

3. Numerical evidence √ P∞ Numerical computation suggests that the series n=1(3 − Ln) indeed converges to L = 4 − ϕ − 2. Pn −455 Denoting Rn := L − k=1(3 − Lk) the presumed remainder, we ﬁnd for instance Rn ≈ 7.34169 × 10 for n = 50 000.

Remark 3.1. Of course, one can also check MUC directly with an algorithm (see e.g. [ Met15 ]). A short Python script took us less than a minute on a personal computer to check MUC for all Markov numbers 1000 mn up to 10 , i.e. up to n = 959 047. Nevertheless, it is nice to get a diﬀerent conﬁrmation. √ 1 C n Pushing the analysis further, we obtain new numerical evidence of Zagier’s estimate mn ∼ 3 e where C = 2.3523414972... . Let us recall that this estimate is still open but was proved in weaker forms

in [ Zag82 ] and [ MR95 ]. Elementary calculus involving the comparison of the remainder Rn with the √ √ √ R +∞ −2C t 6 n −2C n integral 6 n e dt translates Zagier’s estimate to Rn ∼ C e . On Figure 1 it appears that the graph of Rn in Log scale is indeed asymptotic to the expected curve. Remark 3.2 (Computer code). We wrote a simple recursive algorithm in Python to generate the list of Markov numbers. We then used Mathematica to compute the remainders Rn up to n = 50 000 and plot

the graphs. Our code is freely available on GitHub [ js20 ].

10 20 30 40 50 10 000 20 000 30 000 40 000 50 000

10-3 10-100

-200 10-6 10

-300 10-9 10

10-400 10-12

(a) n = 1 ... 50. (b) n = 1 ... 50 000. √ Pn Figure 1. Numerical computation of the remainder Rn = (4−ϕ− 2)− (3−Lk). √ √ k=1 6 n −2C n The dashed curve shows the expected asymptotic proﬁle C e .

References [Aig15] Martin Aigner. Markov’s theorem and 100 years of the uniqueness conjecture. Springer, 2015. [Bow96] B. H. Bowditch. A proof of McShane’s identity via Markoff triples. Bull. London Math. Soc., 28(1):73–78, 1996. [CF89] Thomas W Cusick and Mary E Flahive. The Markoff and Lagrange spectra. Number 30. American Mathematical Soc., 1989. [Coh55] Harvey Cohn. Approach to Markoff’s minimal forms through modular functions. Ann. of Math. (2), 61:1–12, 1955. [Fro13] Georg Ferdinand Frobenius. Über die Markoffschen Zahlen. Königliche Akademie der Wissenschaften, 1913. [Gol03] William M Goldman. The modular group action on real SL(2)–characters of a one-holed torus. Geometry & Topology, 7(1):443–486, 2003. [Gor81] D. S. Gorshkov. Geometry of Lobachevskii in connection with certain questions of arithmetic. Journal of Soviet Mathematics, 16:788–820, 1981. [Guy83] Richard K Guy. Don’t try to solve these problems! The American Mathematical Monthly, 90(1):35–41, 1983.

[js20] jbgaster and seub. GitHub repository: LagrangeSeries, 2020. URL: https://github.com/seub/LagrangeSeries . THE SUM OF LAGRANGE NUMBERS 5

[LT07] Mong Lung Lang and Ser Peow Tan. A simple proof of the Markoff conjecture for prime powers. Geometriae Dedicata, 129(1):15–22, 2007. [Mar79] Andrey Markoff. Sur les formes quadratiques binaires indéfinies. Mathematische Annalen, 15(3-4):381–406, 1879. [Mar80] Andrey Markoff. Sur les formes quadratiques binaires indéfinies II. Mathematische Annalen, 17(3):379–399, 1880. [McS98] Greg McShane. Simple geodesics and a series constant over Teichmüller space. Inventiones Mathematicae, 132(3):607–632, 1998. [Met15] Brandon John Metz. A Comparison of Recent Results on the Unicity Conjecture of the Markoff Equation. Master thesis in Mathematics, University of Nevada, Las Vegas, 2015. [MR95] Greg McShane and Igor Rivin. Simple curves on hyperbolic tori. C. R. Acad. Sci. Paris Sér. I Math., 320(12):1523– 1528, 1995. [Sch96] Paul Schmutz. Systoles of arithmetic surfaces and the Markoff spectrum. Mathematische Annalen, 305(1):191–203, 1996. [Ser85] Caroline Series. The geometry of Markoff numbers. The Mathematical Intelligencer, 7(3):20–29, 1985. [Zag82] Don Zagier. On the number of Markoff numbers below a given bound. Mathematics of Computation, 39(160):709– 723, 1982.

Department of Mathematical Sciences, University of Wisconsin-Milwaukee Email address: [email protected]

Mathematisches Institut, Universität Heidelberg and Heidelberg Institute of Theoretical Studies Email address: [email protected]